The firm has denied misusing data and said it deleted Facebook data after learning the information did not adhere to data protection rules.
Mr Zuckerberg wrote: "We have a responsibility to protect your data, and if we can't then we don't deserve to serve you. I've been working to understand exactly what happened and how to make sure this doesn't happen again.
Mark Zuckerberg announced plans to protect users' data in the future. These include:
Investigating all Facebook apps that had access to 'large amounts' of data before 2014 policy changes came into force
Remove an app's access to your data if you have not used it in three months
Developers will be required to sign a contract in order to ask for access to user posts
Reduce the amount of data it is possible to hand over to an app without explicit permission
Place a tool which easily allows data permissions to be revoked at the top of your news feed
"The good news is that the most important actions to prevent this from happening again today we have already taken years ago.
"But we also made mistakes, there's more to do, and we need to step up and do it.
"I started Facebook, and at the end of the day I'm responsible for what happens on our platform. I'm serious about doing what it takes to protect our community.
"While this specific issue involving Cambridge Analytica should no longer happen with new apps today, that doesn't change what happened in the past.
"We will learn from this experience to secure our platform further and make our community safer for everyone going forward.
"I want to thank all of you who continue to believe in our mission and work to build this community together.
"I know it takes longer to fix all these issues than we'd like, but I promise you we'll work through this and build a better service over the long term."
Facebook shares have slid by more than 7.6 per cent since the first allegations were reported at the weekend by the Observer, and the firm received a backlash online - with a number of users reporting that they were deleting their accounts, including the co-founder of WhatsApp, which was bought by Facebook in 2014.
The company is also facing legal action from some of its own shareholders, who claim the company made "materially false and misleading statements regarding the company's business, operational and compliance policies".
CA chief executive Alexander Nix was suspended after recordings emerged of him making a series of controversial claims, including boasts that CA had a pivotal role in the election of Donald Trump.
The company has denied using Facebook data in its work on the president's election campaign.
Additional reporting by the Press Association